Algebras of quotients of path algebras

Molina, Mercedes Siles

doi:10.1016/j.jalgebra.2007.09.017

Cited by 21 publications

(2 citation statements)

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“…Moreover, at least α j = α * j α j α j appears in the expression for α * j xuα j because if we had α j = α * j α i α j for some i ≠ j, then deg(α i ) = deg(α j ) which implies α * j α i = 0 and therefore α i = 0, a contradiction. This shows that α * j xuα j has at least a nonzero monomial, and because distinct elements of KE are linearly independent (see [22,Lemma 1.1]), then α * j xuα j ≠ 0. Now, this element has at most r summands because α * j α j+1 α j = 0 and it satisfies the induction hypothesis, so that…”

Section: The Prime Spectrum Correspondencementioning

confidence: 96%

“…By using the results above, it is straightforward to check that L(E) is prime but not left (nor right) primitive, whereas L(F ) is left (and right) primitive but not simple. In fact L(E) K[x, x −1 ] (see [1,Examples 1.4]), and L(F ) T where T denotes the algebraic Toeplitz algebra (see [22,Theorem 5.3]).…”

Section: Corollary 310 Let E Be a Graph L(e) Is Prime If And Only If ...mentioning

confidence: 99%

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Prime spectrum and primitive Leavitt path algebras

Pino¹,

Pardo²,

Molina³

2009

Indiana Univ. Math. J.

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Section: The Prime Spectrum Correspondencementioning

confidence: 96%

Section: Corollary 310 Let E Be a Graph L(e) Is Prime If And Only If ...mentioning

confidence: 99%

Prime spectrum and primitive Leavitt path algebras

Pino¹,

Pardo²,

Molina³

2009

Indiana Univ. Math. J.

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Noetherian Leavitt Path Algebras and Their Regular Algebras

Pino

Vaš

2013

Mediterr. J. Math.

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In the past, it has been shown that the Leavitt path algebra L(E) = L K (E) of a graph E over a field K is left and right noetherian if and only if the graph E is finite and no cycle of E has an exit. If Q(E) = Q K (E) denotes the regular algebra over L(E), we prove that these two conditions are further equivalent with any of the following: L(E) contains no infinite set of orthogonal idempotents, L(E) has finite uniform dimension,self-injective and a few more equivalences. In addition, if the involution on the field K is positive definite, these conditions are equivalent with the following: the involution * extends fromis the maximal (total or classical) symmetric ring of quotients of L(E), the maximal right ring of quotients of L(E) is the same as the total (or classical) left ring of quotients of L(E), every finitely generated nonsingular L(E)-module is projective, and the matrix ring M n (L(E)) is strongly Baer for every n. It may not be surprising that a noetherian Leavitt path algebra has these properties, but a more interesting fact is that these properties hold only if a Leavitt path algebra is noetherian (i.e. E is a finite no-exit graph).Using some of these equivalences, we give a specific description of the inverse of the isomorphism V (L(E)) → V (Q(E)) of monoids of equivalence classes of finitely generated projective modules of L(E) and Q(E) for noetherian Leavitt path algebras. We also prove that two noetherian Leavitt path algebras are isomorphic as rings if and only if they are isomorphic as * -algebras. This answers in affirmative the Isomorphism Conjecture for the class of noetherian Leavitt path algebras: if L C (E) and L C (F ) are noetherian Leavitt path algebras, then L C (E) ∼ = L C (F ) as rings implies C * (E) ∼ = C * (F ) as * -algebras.

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Largest Ideals in Leavitt Path Algebras

et al. 2020

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We identify largest ideals in Leavitt path algebras: the largest locally left/right artinian (which is the largest semisimple one), the largest locally left/right noetherian without minimal idempotents, the largest exchange, and the largest purely infinite. This last ideal is described as a direct sum of purely infinite simple pieces plus purely infinite non-simple and non-decomposable pieces. The invariance under ring isomorphisms of these ideals is also studied.

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Algebras of quotients of path algebras

Cited by 21 publications

References 20 publications

Prime spectrum and primitive Leavitt path algebras

Prime spectrum and primitive Leavitt path algebras

Noetherian Leavitt Path Algebras and Their Regular Algebras

Largest Ideals in Leavitt Path Algebras

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